Abstract
The flux reconstruction (FR) methodology has proved to be an attractive approach to obtaining high-order solutions to hyperbolic partial differential equations. However, the utilization of somewhat arbitrarily defined correction polynomials in the application of these schemes, while adding some flexibility, detracts from their ease of implementation and computational efficiency. This paper describes a simplified formuation of the flux reconstruction method that replaces the application of correction polynomials with a single Lagrange interpolation operation. A proof of the algebraic equivalence of this scheme to the FR formulation of the nodal discontinuous Galerkin (DG) method provided that the interior solution points are placed at the zeros of a corresponding Legendre polynomial is presented. Next, a proof of linear stability for this formulation is given. Subsequently, von Neumann analysis is carried out on the new formulation to identify a range of linearly stable schemes achieved by variations of the interior solution point locations. This analysis leads to the discovery of linearly stable schemes with greater formal order of accuracy than the DG method.
Similar content being viewed by others
References
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749 (2001)
Asthana, K., Jameson, A.: High-order Flux Reconstruction Schemes with Minimal Dispersion and Dissipation. J. Sci. Comput. 62, 913 (2015)
Cockburn, B., Shu, C.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173 (2001)
Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids. J. Comput. Phys. 181, 186 (2002)
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods- Algorithms, Analysis, and Applications. Springer, Berlin (2008)
Hildebrand, F.B.: Introduction to Numerical Analysis. Dover, New York (1974)
Huynh, H.T.: A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods, AIAA Conference Paper, 2007, p 4079 (2007)
Jameson, A.: A Proof of the stability of the spectral difference method for all orders of accuracy. J. Sci. Comput. 45(1–3), 348–358 (2010)
Kopriva, D.A., Kolias, J.H.: A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125, 244 (1996)
Liu, Y., Vinokur, M., Wang, Z.J.: Spectral difference method for unstructured grids I: basic formulation. J. Comput. Phys. 216, 780 (2006)
Reed, W.H., Hill, T.R.: Triangular Mesh Methods for the Neturon Rransport Equation. Technical Report LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, New Mexico, USA (1973)
Rusanov, V.V.: Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. 1, 267 (1961)
Vincent, P.E., Castonguay, P., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput. 47(1), 5072 (2011)
Vincent, P.E., Castonguay, P., Jameson, A.: Insights from von Neumann analysis of high-order flux reconstruction. J. Comput. Phys. 230, 81348154 (2011)
Wolfram Research, Inc., Mathematica, Version 10.1, Champaign, IL (2015)
Acknowledgments
The first author would like to acknowledge support from the Morgridge Family Stanford Graduate Fellowship. The second author would like to acknowledge support from the Thomas V. Jones Stanford Graduate Fellowship.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Romero, J., Asthana, K. & Jameson, A. A Simplified Formulation of the Flux Reconstruction Method. J Sci Comput 67, 351–374 (2016). https://doi.org/10.1007/s10915-015-0085-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-015-0085-5